Universal Sequence Preconditioning

TL;DR

This paper introduces a universal preconditioning method for sequential prediction that improves regret bounds and enhances the performance of various algorithms, including neural networks, by convolving target sequences with orthogonal polynomial coefficients.

Contribution

It proposes a novel preconditioning technique based on polynomial convolution that achieves sublinear, dimension-independent regret bounds for linear dynamical systems.

Findings

Reduces regret for multiple prediction algorithms.

Improves performance of recurrent neural networks.

Effective on both synthetic and real-world data.

Abstract

We study the problem of preconditioning in sequential prediction. From the theoretical lens of linear dynamical systems, we show that convolving the target sequence corresponds to applying a polynomial to the hidden transition matrix. Building on this insight, we propose a universal preconditioning method that convolves the target with coefficients from orthogonal polynomials such as Chebyshev or Legendre. We prove that this approach reduces regret for two distinct prediction algorithms and yields the first ever sublinear and hidden-dimension-independent regret bounds (up to logarithmic factors) that hold for systems with marginally table and asymmetric transition matrices. Finally, extensive synthetic and real-world experiments show that this simple preconditioning strategy improves the performance of a diverse range of algorithms, including recurrent neural networks, and generalizes to signals beyond linear dynamical systems.